There are various strict monoidal model categories of spectra (e.g. symmetric spectra) where the honestly commutative monoid objects model the "coherently commutative" ring spectra (which might otherwise be expressed using, say, operads). Is there an analog for spaces? That is, there a monoidal model category, Quillen equivalent to spaces (in some monoidal sense), such that the category of commutative monoids in this category is (Quillen) equivalent to the category of algebras in spaces over some fixed and suitably free $E_\infty$-operad? In spaces, this is false; topological abelian groups are very far from modelling infinite loop spaces.
Yes, such a model is developed in a paper of Blumberg, Cohen and Schlichtkrull about Thom spectra.